Is there a rule of thumb for small angle approximation?

[Starwatcher16]

When you are not given an acceptable level of error in a problem, is there any rule of thumb I should use for how large Theta can be before I stop using the small angle approximation(Sin Theta=Theta) ?

 

[rock.freak667]

According to my textbook it says (to 3sf, in radians)

$$-0.105<\theta<0.105$$

 

[Hurkyl]

When you are not given an acceptable level of error in a problem, is there any rule of thumb I should use for how large Theta can be before I stop using the small angle approximation(Sin Theta=Theta) ?

Decide for yourself what level of error is appropriate. Then use your calculus to determine if that approximation is good enough!

The Taylor remainder theorem is a systematic way to bound the error on an approximation.

However, when theta is small, the Taylor series for sin is an alternating series whose terms are strictly decreasing -- so an easier method is to use what you know about alternating series to estimate the error.

 

[protonchain]

In general if you're doing really simple and very very approximate calculations then anything less than 0.5 would probably be just fine.

Truly though, the best way to look is to see a graph of Sin x / x. Analyze this and look at the regions where it is 0.95 - 1 for example, and you can then grasp where you can begin to approximate.

About x = 0.55 is where Sin x / x = 0.95. Obviously as x goes to 0, the value increases towards 1.

I hope that helped or at least made you get up and get your graphing calculator :P